We propose an efficient computation method for the infinite integral integral(infinity)(0) xdx/(1 + x(6) sin(2) x), whose integrand contains a series of spikes, approximately pi apart, growing taller and narrower as x increases. Computing the value of this integral has been a problem since 1984. We herein demonstrate a method using the Hilbert transform for changing this type of singular function into a smooth function and computing the value of the integral to more than one million significant digits using a superconvergent double exponential quadrature method. (C) 2013 Elsevier B.V. All rights reserved.